How to make a H1 different then the Title?

Yes! Custom fields: http://codex.wordpress.org/Custom_Fields

You can create alternate post titles via the post editor, and then you can display them with some minor adjustments to the single.php and index.php templates (or {$post_type}-archive.php templates, if you have them). For example:

<?php $post_title = get_post_meta($post->ID, 'Post-Title', true); //"Post-Title" or whatever you call your custom field
if ($post_title) {
?>
    <h1><?php echo $post_title; ?></h1>

<?php } else { ?>

    <h1><?php the_title(); ?></h1>

<?php } ?>

Alex Denning has a great introduction on Smashing Magazine here: http://wp.smashingmagazine.com/2010/04/29/extend-wordpress-with-custom-fields/

EDIT

If you’d like to make this update-proof, I would suggest making a handy little plugin. For something to try, take the code below and replace “YOUR_CUSTOM_FIELD_HERE” with your custom field, and then it will filter the template tag the_title(); to replace it with that field (if not filled in, returns the post title). I haven’t tested this code, but it should work:

<?php
/**
 * Plugin Name: Replace Title With Custom Field
 * Description: When activated, this plugin replaces the title H1 with a custom field from the dashboard.
 * Version: 1.0
 * Author: penguin429
 * License: GPL2
 */

function replace_title($title, $id) {
    $custom_title = get_post_meta($id, 'YOUR_CUSTOM_FIELD_HERE', true);
    $mod_title = $title;

    if(!empty($custom_title)) { //if field is filled in, make the title the Custom Field
        $mod_title = $custom_title;
    } else { //otherwise, return the post Title
        $mod_title = $title;
    }
    return $mod_title;
}
add_filter('the_title', 'replace_title', 10, 2);

?>

To try it out, take the code and give it a name (say, custom-title.php) and drop it in your site’s wp-content/plugins folder and activate it. Let me know if it works out!